TY - JOUR

T1 - Bernoulli convolutions and an intermediate value theorem for entropies of K-partitions

AU - Lindenstrauss, Elon

AU - Peres, Yuval

AU - Schlag, Wilhelm

PY - 2002

Y1 - 2002

N2 - We establish a strong regularity property for the distributions of the random sums ∑ ± λn, known as "infinite Bernoulli convolutions": For a.e. λ ∈ (1/2, 1) and any fixed l, the conditional distribution of (ωn+1, ωn+l) given the sum ∑n=0 ∞= ωnλn, tends to the uniform distribution on (±1)l as n → ∞. More precise results, where l grows linearly in n, and extensions to other random sums are also obtained. As a corollary, we show that a Bernoulli measure-preserving system of entropy h has K-partitions of any prescribed conditional entropy in [0, h]. This answers a question of Rokhlin and Sinai from the 1960's, for the case of Bernoulli systems.

AB - We establish a strong regularity property for the distributions of the random sums ∑ ± λn, known as "infinite Bernoulli convolutions": For a.e. λ ∈ (1/2, 1) and any fixed l, the conditional distribution of (ωn+1, ωn+l) given the sum ∑n=0 ∞= ωnλn, tends to the uniform distribution on (±1)l as n → ∞. More precise results, where l grows linearly in n, and extensions to other random sums are also obtained. As a corollary, we show that a Bernoulli measure-preserving system of entropy h has K-partitions of any prescribed conditional entropy in [0, h]. This answers a question of Rokhlin and Sinai from the 1960's, for the case of Bernoulli systems.

UR - http://www.scopus.com/inward/record.url?scp=0036436442&partnerID=8YFLogxK

U2 - 10.1007/BF02868480

DO - 10.1007/BF02868480

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AN - SCOPUS:0036436442

SN - 0021-7670

VL - 87

SP - 337

EP - 367

JO - Journal d'Analyse Mathematique

JF - Journal d'Analyse Mathematique

ER -